LPTHE Orsay 98/64
November 1998
arXiv:hep-ph/9811346v3 13 Jan 1999
THE LIKELIHOOD OF DCC FORMATION
André KRZYWICKI and Julien SERREAU
LPTHE, Bâtiment 211, Université Paris-Sud, 91405 Orsay, France1
Abstract
We estimate the probability that a disoriented chiral condensate
forms during the spherical expansion of a hot medium described by the
linear sigma model.
1
Introduction
The suggestion [1]-[4] that disoriented chiral condensates (DCC) could form in
high-energy hadronic collisions has attracted much attention. More than hundred publications are devoted to the study of various aspects of this hypothetical phenomenon (see e.g. the reviews [5, 6]). The most plausible mechanism of
DCC formation has been identified: fast expansion of hot quark-gluon plasma
results in a rapid suppression of thermal fluctuations (quenching), which in
turn triggers a dramatic amplification of soft pion modes (see e.g. refs. [7, 8, 9].
However, nobody made a serious attempt to estimate the probability that this
happens. It is true that the present models are not realistic enough to be
trusted at the quantitative level. Nevertheless, even within the existing framework it is legitimate to seek for a crude estimate of the probability in question.
This is the problem we address in the present paper.
It is expected that the chiral symmetry is restored in a sufficiently hot
quark-gluon plasma and that it is spontaneously broken when the plasma is
cooled. At the same time, freely propagating quarks and gluons get trapped
into hadrons. It would be too difficult to model both phenomena, symmetry
breaking and confinement. Hence, in the DCC literature it is customary to
describe the soft modes of the cooling medium by the linear sigma model,
evacuating cavalierly the confinement problem. The justification of this idealization, which we also adopt, is discussed at length in many places.
1
Laboratoire associé au Centre National de la Recherche Scientifique - URA00063.
1
The fast expansion can be
q modeled by assuming that the field depends
on time via the variable τ = t2 − xµ xµ , the index µ running from 1 to D.
In the equations of motion the operator ∂ 2 /∂t2 is then replaced by ∂ 2 /∂τ 2 +
(D/τ )∂/∂τ . In addition to the acceleration term, there also appears a ”friction” term reflecting the decrease of energy in a covolume [3]. Since the ”friction force” is proportional to D, the larger is D the more efficient is the
quenching [9].
In this paper we focus on the spherical geometry, the most favourable
one for DCC formation. We imagine a droplet of DCC undergoing a radial
expansion in its rest frame. As remarked in [10], where the same geometry is
studied, this does not necessarily mean that the full collision process has the
same symmetry.
The time evolution of the chiral phase transition during a spherical expansion of a medium described by the linear sigma model was studied in ref.
[8]. We adopt most of their formalism here and, in particular, the two basic
ingredients: the assumption that the system is initially in local thermal equilibrium and the self-consistent approximation of the mean-field type. We shall
argue that essentially no other assumptions are needed to determine the probability that soft modes are amplified by a given amount. This is, however, not
quite enough to fix the probability of observing a DCC in a collision process.
Information on the environement of the DCC bubble and on the experimental
set-up is needed for that. We shall come back to this question towards the
end of this paper.
In the next section we review briefly the formalism. In sect. 3 we explain
our sampling strategy for initial conditions. The results and discussion are in
sect. 4.
2
The formalism
Let us briefly summarize the formalism of ref. [8]. The authors start with the
linear sigma model for an N-component scalar field (eventually N is set to 4).
Using the 1/N approximation they derive the following dynamical equations
of the Hartree type:
(∂ 2 + χ)φ̄(x) = Hnσ ,
(∂ 2 + χ)φ(x) = 0,
(1)
(2)
where χ satisfies the gap equation
χ = λ0 (φ̄ 2 + hφ2 i − v02 ),
(3)
φ̄ is the classical mean-field and φ is the quantum fluctuation. The parameters
λ0 and v0 are the bare couplings of the sigma model. An explicit symmetry
2
breaking term, proportional to H, gives a finite mass to the Goldstone boson.
Of course, hφ2 i diverges and it is necessary to set an ultra-violet cut-off Λ.
The quadratic divergence is removed by subtracting the mass gap at zero
temperature, T = 0, obtaining
1
(χ − m2π ) = φ̄ 2 − fπ2 + hφ2 i − hφ2 iT =0
λ0
(4)
The remaining logarithmic divergence is eliminated by introducing the renormalized coupling constant:
−1
2
λ−1
r = λ0 + (N/8π )
Z
0
Λ
k 2 dk/(k 2 + m2π )3/2
(5)
The cancellation of divergences is, at any time, insured by an appropriate
quantization condition (cf eqs. (11) below). The theory becomes trivial when
Λ → ∞ and therefore the cut-off has to be kept finite. The effect of the
renormalization is to reduce the cut-off dependence of the results.
Notice, that in this approximation, unlike in the standard Hartree one,
√
all components of the fluctuation field have the same mass χ, whose value
is independent of the orientation of the order parameter φ̄. This is not very
realistic but is acceptable if one wishes to estimate only the amplification of
the length of the field vector, without paying attention to its orientation in
the internal space.
In order to take care of the expansion, the time t and the radial coordinate
r are replaced by the hyperbolic coordinates
√
(6)
τ = t2 − r 2 , η = tanh−1 (r/t)
and the theory is quantized on hypersurfaces τ = const. It is assumed that
the mean-field depends on τ only. Projecting the fluctuation field on the
orthonormal eigenmodes Ys (η, θ, ϕ) of the curved Laplacian (cf [11]; we use
the shorthand notation s = (s, l, m), s is dimensionless ) one gets
φs (τ ) = ψs (τ )as + (−1)m ψ̄s (τ )a†-s
φ̇s (τ ) = ψ̇s (τ )as + (−1)m ψ̄˙ s (τ )a†-s
(8)
[ais , a†js’ ] = δij δss’
(9)
where
(7)
and the dot represents the differentiation with respect to τ . The problem can
be reduced to the familiar one of a set of parametrically excited harmonic
oscillators. For that purpose one replaces ψs → gs /τ and one introduces the
”time” variable u = log (mπ τ ) to get
[
d2
+ ωs (u)]gs (u) = 0
du2
3
(10)
q
where ωs (u) = s2 + χ(τ )τ 2 is the dimensionless ”frequency”. The oscillators
are coupled through the common mass. The physical frequency is ωs (u)/τ .
The quantization is completed by choosing appropriate initial conditions
for the mode functions. The cancellation of divergences required by the renormalization of the mass gap is insured if one adopts the following adiabatic
condition2 at u = u0:
gs (u0 ) = gs(0) (u0 ), gs′ (u0 ) = gs(1) (u0)
(11)
with
q
gs(0) (u) = 1/ 2ωs (u)
(12)
gs(1) (u) = −[ωs′ (u)/2ωs (u) + iωs (u)]gs(0) (u)
(13)
the prime representing the differentiation with respect to u.
Assume that Heisenberg and Schroedinger representations coincide at
τ = τ0 . One can then regard a†s as the Schroedinger representation operator
creating a particle with frequency ωs (u0 )/τ0 . These are the particles appropriate for the description of the initial state. One can introduce analogously
the particles with frequencies ωs (uf )/τf appropriate to the final state and
counted at τ = τf . Let b†s denote the corresponding creation operator, in the
Schroedinger representation. The Heisenberg representation field operators,
given by (7)-(8), can then also be written
φs (τ ) = ψs(0) (τf )bs (τ ) + (−1)m ψ̄s(0) (τf )b†-s (τ )
φ̇s (τ ) = ψs(1) (τf )bs (τ ) + (−1)m ψ̄s(1) (τf )b†-s (τ )
(14)
(15)
where ψs(0) (τ ) = gs(0) (u)/τ and ψs(1) (τ ) = [gs(1) (u) − gs(0) (u)]/τ 2 while bs (τ ) =
U(τ, τ0 )bs U † (τ, τ0 ), U being the unitary operator connecting the two representations (when the potential is quadratic U can be explicitly constructed
[13], but we do not need this construction here3 ). Using (7)-(8) and (14)-(15)
one easily finds the Bogoliubov transformation connecting the operators as , a†s
and bs (τ ), b†s (τ ):
bs (τ ) = α(τ )as + β(τ )(−1)m a†-s
(16)
where
2
αs (τ ) = i[gs′ (u)ḡs(0) (uf ) − gs (u)ḡs(1) (uf )]
βs (τ ) = i[ḡs′ (u)ḡs(0) (uf ) − ḡs (u)ḡs(1) (uf )]
(17)
(18)
Only second order adiabatic condition is imposed. This may be insufficient in the general
context [12] but is enough when one only seeks to insure the proper renormalization of the
mass gap.
3
The operator U combines free propagation and squeezing. The latter takes care of the
creation of particle pairs.
4
Assuming
(a)
ha†is aj s’ i = nis (τ0 )δij δss’ ,
hais aj s’ i = 0
(19)
one finally obtains4 , for one field component, the number of b-particles at
time τ for a given number of a-particles at time τ0 5 :
(b)
nis (τ ) +
1
1
(a)
= As (τ )[nis (τ0 ) + ]
2
2
(20)
where
As = 2 | βs |2 +1
(21)
The multiplicity ns(b) (τ ) calculated at τ = τf depends weakly on the choice of
the reference final time τf , when the latter is large. Assuming local thermal
equilibrium in the initial state one sets ns(a) (τ0 ) = {exp [ωs (u0 )/τ0 T ] − 1}−1 .
3
Physical picture and the sampling strategy
As already mentioned in the Introduction, we consider the evolution of a spherical droplet of DCC in its rest frame. We start with a small ball of radius R,
filled with hot matter in local thermal equilibrium. We assume that the ball
expands at the speed of light. Due to the time dilation the temperature and
the value of the mean-field stay approximately constant within a layer near the
boundary of the ball. Of course, the width of this layer shrinks to zero with
increasing time (this width equals the distance between the surface τ = R and
the light cone). The equations of motion of the sigma model are supposed to
describe what happens in the interior of the ball, the cooling observed as one
moves away from the surface towards the center. We make first the unrealistic
assumption that the DCC is connected forever to a heat bath kept at constant
temperature, so that the process never stops. Later on we shall discuss the
effect of switching the heat bath off.
The evolution of the droplet depends on the initial conditions set at
τ = τ0 = R. It is easy to convince oneself that in the Hartree approximation
the only relevant initial conditions are those concerning the mean-field and
its time derivative (the fluctuations of the initial occupancy numbers appear
4
The derivation given here differs from that of ref. [8], where the so-called adiabatic
basis is unnecessarily used. Strictly speaking, the Bogoliubov transformation given by their
eqs. (3.19) is not unitary when the frequency is imaginary in some time interval, since
the exponential factor entering the definition of the adiabatic basis is then no longer a pure
phase. We understand that in actual calculations the amplification factor was obtained from
the formula derived for real frequencies. We reproduce their results using eqs. (17)-(18).
5
J. Randrup has emphasized that the amplification of vacuum fluctuations contributes
significantly to the effect, which is therefore likely to be underestimated using classical
equations of motion [14]
5
only in the formula for the initial mass gap and approximately average to zero
there). How to choose these parameters? In all publications up to now they
were given values ad hoc. We shall argue, that the probability that φ̄(τ0 ) and
φ̄˙ (τ0 ) take a given value is determined once one has assumed that the droplet
is initially in local thermal equilibrium. In order to make the point let us for
a moment neglect the complications due to quantum mechanics and let us us
consider a classical field.
Consider a ball of fixed volume V0 filled with the field in local thermal
equilibrium. Hence, the field fluctuates as if the ball were part of a system
in equilibrium, with volume V much larger than V0 . In this large system the
variances of the spacial averages of the field and of its time derivative would
be very small, of order 1/V (since long range correlations are absent). The
corresponding variances for averages calculated by integrating over the volume
of the ball would be larger by a factor of order V /V0 (assuming that the radius
of the ball is at least of the order of the correlation length in the medium).
An observer living within the ball would identify the spatial averages of the
field and of its time derivative with φ̄(τ0 ) and φ̄˙ (τ0 ), respectively. The point
is that these random variables are fluctuating in a predictable manner.
In quantum theory φ̄(τ0 ) and φ̄˙ (τ0 ) can be sampled from the probability
distribution6 characterized by the mean
E[φ̄i (τ0 )] = δi0 H/χT ,
E[φ̄˙ i (τ0 )] = 0,
(22)
(23)
where χT is the mass gap (squared) in equilibrium at temperature T , by the
variances
1
d3 x d3 x′ hφi(x)φi (x′ )i
V02 V0
Z
1
˙
Var[φ̄i (τ0 )] =
d3 x d3 x′ hφ̇i(x)φ̇i (x′ )i
V02 V0
Var[φ̄i (τ0 )] =
Z
(24)
(25)
and by the covariance
1
Cov[φ̄i (τ0 ), φ̄˙ (τ0 )] = 2
V0
Z
1
d3 x d3 x′ h [φi (x)φ̇i (x′ ) + φ̇i (x′ )φi (x)]i
2
V0
(26)
In the Hartree approximation the interacting system is replaced by an ensemble
√
of free excitations, with mass χT . It is not difficult to convince oneself that in
this approximation and in the thermal canonical ensemble with density matrix
∝ exp(−H0 /T ), where H0 is the free Hamiltonian, the probability distribution
6
The probability distribution of the measured values of an observable O is determined
by the characteristic function heikO i.
6
we are interested in is Gaussian. Furthermore the covariance vanishes. Hence
the parameters given by (22)-(25) determine the distribution, actually the
analogue of the Wigner function, exactly.
The variances (24)-(25) can be estimated analytically, when the radius R
√
of the ball is much larger than the correlation length λ = 1/ χT . It is sufficient
to calculate the variances for the quasi-infinite volume V and to multiply the
result by V /V0 , since the V /V0 small cells fluctuate independently. One gets
√
3 coth ( χT /2T )
Var[φ̄i (τ0 )] =
(27)
√
8π χT R3
√
√
3 χT coth ( χT /2T )
˙
(28)
Var[φ̄i (τ0 )] =
8πR3
The dispersion of φ̄i calculated exactly is smaller by a factor of 2 (1.5) for
R = λ (R = 2λ) and T = 200 to 400 MeV. The dispersion of φ̄˙i obtained from
(28) is off by 20% (8% ), respectively. For R < λ the disrepancy between the
analytical formulae and the exact results increases rapidly. Thus, as expected,
the fluctuations within the ball depend rather weakly on the environement
provided R >
∼ λ.
The formalism of ref. [8], reviewed in sect. 2, together with the sampling
method proposed in this section enable one to estimate the likelihood of a
coherent amplification of the pion field. More precisely, one can calculate the
probability that the amplification factor As given by eq. (21) takes a given
value. In such a calculation the size of the initial ball, viz. R, appears as a
free parameter. Remember, however, that one has to set τ0 = R and that
the friction force responsible for the quench is proportional to 1/τ . Thus
the likelihood of DCC formation decreases with increasing R: this parameter
should be assigned the smallest possible value in order to get the estimate we
are looking for.
At this point of the discussion let us remark that the theory we wish
to use makes only sense for R >
∼ λ. Indeed, the concept of local thermal
equilibrium is meaningfull when applied to a cell whose degrees of freedom
fluctuate more or less independently from what happens in the neighbor cells.
Also, the validity of the Hartree approximation requires the size of the cell to be
larger than the Compton wavelength of an excitation. With these arguments
in mind we focus on the values of R in the range of one to two correlation
lengths 7 .
7
We cannot exclude that a DCC instability develops in a smaller cell, but we have no
theory to deal with such a scenario.
7
4
Results and discussion
We show in Fig. 1 the histograms of the probability P (A) that the amplification factor of the s = 0+ mode is A0 ≡ A . It was convenient to assign to the
parameters of the model the values already used in ref. [8]8 . Proceeding in this
way we could check our results against theirs. The amplification is calculated
at τ = τf = 10fm, where the system is in the stationary regime. It is seen
from the figure that P (A) falls rapidly with A. Clearly, large amplification
occurs in a small fraction of events only, especially with the choice R = 2λ.
This is qualitatively similar to the analytical result obtained in ref. [3] using a
1+1 dimensional toy model, except that in the present case P (A) has a power
falling tail at large A:
The histogram corresponding to T = 200 MeV and R = λ is fairly well
represented by
0.805
P (A) =
(29)
(1 + 0.27A)3.21
The corresponding fit for T = 400 MeV is
P (A) =
0.213
(1 + 0.032A)6.959
(30)
It is impossible to find a good fit reproducing the tail of the histogram with
an exponential function, e.g. with P (A) ∝ exp (−aAb ).
It is interesting to inquire what characterizes the initial states leading
to large amplifications. It appears that one condition is the smallness of the
absolute strength of the initial classical isovector current (calculated using
φ̄(τ0 ) and φ̄˙ (τ0 )). This is ilustrated in Fig. 2, where we show the square of
the isovector (resp. isoaxialvector) current versus the amplification factor A.
In order to judge what amplification should be considered as ”large” one
has to estimate the multiplicity of produced pions. This can be done with the
help of the formula (cf [15]) :
E
dn
=
d3 p
Z
√
d4 x −gδ(τ − τf )f (x, p)pµ uµ
(31)
where f (x, p) is the invariant phase-space density and uµ = xµ /τ is a unit 4vector orthogonal to the hypersurface τ = τf , where the particles are counted9 .
Eq. (31) becomes identical to the eq. (B2) used for the same purpose in ref.
8
Hence, Λ = 800 MeV, λr = 7.3, fπ = 92.5 MeV. We find the correlation length λ = 1.17
fm (0.68 fm) at T = 200 Mev (400 MeV).
9
The textbook formula for the intensity of the black-body radiation is obtained replacing
in (31) the constraint τ = τf by t =const, so that pµ uµ = E, and by setting f (x, p) =
2(2π)−3 [exp (E/T ) − 1]−1 . The factor 2 is the number of photon polarization states. This
example helps fixing the normalizations.
8
10
10
10
10
10
−2
10
P(A)
P(A)
10
(a)
0
−4
10
−6
10
−8
0.0
50.0
10
100.0
(b)
0
−2
−4
−6
−8
0.0
50.0
A
10
10
10
10
10
−2
10
P(A)
P(A)
10
(c)
0
−4
10
−6
10
−8
0.0
10.0
100.0
A
10
20.0
(d)
0
−2
−4
−6
−8
0.0
10.0
A
20.0
A
Figure 1: The amplification factor A of the softest mode for (a) T = 200MeV and
R = λ (3.2 × 104 MC events) , (b) T = 400MeV and R = λ (104 MC events) , (c)
T = 200MeV and R = 2λ (2.5 × 104 MC events) , (d) T = 400MeV and R = 2λ
(104 MC events) .
[8], when one goes over to the hyperbolic coordinates (6) and integrates out
the delta function10 . We further set
f (x, p) = N(2π)−3 n(b)
s
(32)
and, as in [8], we relate s to the 4-momentum pµ by the obvious relation
pµ u µ =
q
(s/τf )2 + m2π
(33)
The integrand in (31) depends on a single external 4-vector, viz. pµ , and
10
Except for a factor 2π, coming from the integration over the azimuthal angle, which is
missing in (B2) due to a typo.
9
vector2
0.1
0.0
0.0
50.0
100.0
150.0
100.0
150.0
axial2
A
0.2
0.1
0.0
0.0
50.0
A
Figure 2: Vµ · Vµ [resp. Aµ · Aµ ], the square of the initial classical isovector [resp.
isoaxialvector ] current (in fm−6 ) versus the amplification factor A. We use Monte
Carlo data corresponding to R = λ and T = 200 MeV (3.2 × 104 MC events).
therefore the invariant spectrum is flat11 :
E
dn
= cA
d3 p
(34)
The constant c on the rhs is varying from one Monte Carlo event to
another, but for A > 30 its average is roughly c = 5GeV−2 for R = λ and both
T = 200 MeV and T = 400 MeV. The flatness of the spectrum is, of course,
an artifact of the unrealistic assumption that the boost invariant expansion
continues forever. It is worth mentioning that the rhs of (34) does not depend
on the choice of τf , provided the input f (x, p) in (31) is a time independent
function of s.
11
We are puzzled by the falling DCC spectrum shown in fig. 7 of ref. [8].
10
At this point one should distinguish between the intrisic DCC dynamics
and the extrinsic aspects of DCC formation, those determined by the behavior
of the environement of the DCC bubble. A discussion of the latter, which
would require the modeling of the collision proces as a whole, is beyond the
scope of this paper. Let us only remark that in a real collision process the
expansion would last a finite time. The resulting spectrum would be cut, the
value of the cut-off reflecting the behavior of the environement. Unfortunately,
the predicted total multiplicity depends strongly on this cut-off and cannot
be estimated in a reliable manner. However, the rhs of (34) is presumably
a reasonable estimate of the invariant momentum space density of soft pion
radiation.
A simple example is instructive: Denoting by h the height of the rapidity
”plateau”, one can parametrize the one-particle spectrum in the central rapidp2t
h
ity region writing E ddn
3 p = πhp2 i exp (− hp2 i ). In a central Pb-Pb colision at the
t
t
CERN SPS one observes [16] about 200 π − per unit rapidity, i.e. for all pions
h ≈ 600. Thus the invariant momentum space density at very small transverse
momentum is roughly 1900 GeV−2 , where we have used hp2t i = 0.1GeV2 . The
corresponding density fluctuation is expected to be of the order of 75 GeV−2 .
The rhs of (34) should be significantly larger than the fluctuation, for the DCC
signal to be detectable12 . The signal would be more than three times the fluctuation if A > 45. Setting R = λ, the corresponding probability is roughly
4 × 10−3 . Of course, this is a conditional probability, as it has been assumed
that the initial plasma droplet was formed in the collision13 .
Summarizing: The mean-field approximation of ref.[8] is combined with
a specific sampling method designed to generate the initial values taken by
the mean-field and its time derivative. The method rests on the assumption
that the medium is initially in a state of local thermal equilibrium. The
long-wavelength excitations of the medium are modeled with the linear sigma
model. The probability that the amplification of the soft modes has a given
magnitude has been estimated. The probability of an observable DCC signal
appears small. A crude estimate indicates that in a central Pb-Pb collision at
CERN SPS this probability is at best of the order of 10−3 . This indication
should be taken into account by experimenters designing DCC hunt strategies.
Acknowledgements: We acknowledge useful conversations/correspondence with F. Cooper, B. Jancovici, J. Jurkiewicz, M.A. Lampert, A.H. Mueller, J.Y. Ollitraut, R. Omnes and J. Randrup.
12
Hopefully, a DCC signal can be distinguished from a large fluctuation due to its specific
charge structure.
13
We get almost the same estimate at 200 and 400 MeV. Although at 200 MeV the initial
fluctuations are smaller, the friction is weaker, since we take a larger initial ball.
11
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12